L(s) = 1 | + 2.30·2-s + 3-s + 3.30·4-s − 2.10·5-s + 2.30·6-s − 7-s + 3.01·8-s + 9-s − 4.85·10-s − 1.64·11-s + 3.30·12-s − 0.138·13-s − 2.30·14-s − 2.10·15-s + 0.331·16-s + 4.32·17-s + 2.30·18-s + 1.02·19-s − 6.96·20-s − 21-s − 3.79·22-s − 9.26·23-s + 3.01·24-s − 0.568·25-s − 0.318·26-s + 27-s − 3.30·28-s + ⋯ |
L(s) = 1 | + 1.62·2-s + 0.577·3-s + 1.65·4-s − 0.941·5-s + 0.940·6-s − 0.377·7-s + 1.06·8-s + 0.333·9-s − 1.53·10-s − 0.496·11-s + 0.955·12-s − 0.0383·13-s − 0.615·14-s − 0.543·15-s + 0.0828·16-s + 1.04·17-s + 0.543·18-s + 0.235·19-s − 1.55·20-s − 0.218·21-s − 0.808·22-s − 1.93·23-s + 0.615·24-s − 0.113·25-s − 0.0625·26-s + 0.192·27-s − 0.625·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 5 | \( 1 + 2.10T + 5T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 + 0.138T + 13T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 - 1.02T + 19T^{2} \) |
| 23 | \( 1 + 9.26T + 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 31 | \( 1 + 3.66T + 31T^{2} \) |
| 37 | \( 1 - 0.470T + 37T^{2} \) |
| 41 | \( 1 + 9.09T + 41T^{2} \) |
| 43 | \( 1 - 8.22T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 2.05T + 53T^{2} \) |
| 59 | \( 1 + 1.22T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 + 3.48T + 71T^{2} \) |
| 73 | \( 1 - 2.56T + 73T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 8.81T + 89T^{2} \) |
| 97 | \( 1 + 4.31T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.40244175737357723490266731767, −6.71312609695294506538234256665, −5.87284839893810134651172904078, −5.34910711887613969027816258170, −4.43926791497295338323132607736, −3.87273343719794668132003946399, −3.35660615754912044715626169379, −2.67926426170625486927353286256, −1.71274134612430171393519342193, 0,
1.71274134612430171393519342193, 2.67926426170625486927353286256, 3.35660615754912044715626169379, 3.87273343719794668132003946399, 4.43926791497295338323132607736, 5.34910711887613969027816258170, 5.87284839893810134651172904078, 6.71312609695294506538234256665, 7.40244175737357723490266731767